display all the ideas for this combination of texts
5 ideas
| 16918 | Mathematics cannot proceed just by the analysis of concepts [Kant] |
| Full Idea: Mathematics cannot proceed analytically, namely by analysis of concepts, but only synthetically. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: I'm with Kant insofar as I take mathematics to be about the world, no matter how rarefied and 'abstract' it may become. |
| 16930 | Geometry is not analytic, because a line's being 'straight' is a quality [Kant] |
| Full Idea: No principle of pure geometry is analytic. That the straight line beween two points is the shortest is a synthetic proposition. For my concept of straight contains nothing of quantity but only of quality. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269) | |
| A reaction: I'm not sure what his authority is for calling straightness a quality rather than a quantity, given that it can be expressed quantitatively. It is a very nice example for focusing our questions about the nature of geometry. I can't decide. |
| 16919 | Geometry rests on our intuition of space [Kant] |
| Full Idea: Geometry is grounded on the pure intuition of space. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: I have the impression that recent thinkers are coming round to this idea, having attempted purely algebraic or logical accounts of geometry. |
| 16920 | Numbers are formed by addition of units in time [Kant] |
| Full Idea: Arithmetic forms its own concepts of numbers by successive addition of units in time. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: It is hard to imagine any modern philosopher of mathematics embracing this idea. It sounds as if Kant thinks counting is the foundation of arithmetic, which I quite like. |
| 16929 | 7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant] |
| Full Idea: The concept of twelve is in no way already thought by merely thinking the unification of seven and five, and though I analyse my concept of such a possible sum as long as I please, I shall never find twelve in it. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269) | |
| A reaction: It might be more plausible to claim that an analysis of 12 would reveal the concept of 7+5. Doesn't the concept of two collections of objects contain the concept of their combined cardinality? |